Verify the identity.

tangent left parenthesis x plus StartFraction pi Over 2 EndFraction right parenthesistanx+π2equals=negative cotangent x−cotx

We are asked to verify the identity:

\[
\tan \left( x + \frac{\pi}{2} \right) = -\cot(x)
\]

### Step-by-step verification:

1. **Use the tangent addition formula**:
   \[
   \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}
   \]
   For \(a = x\) and \(b = \frac{\pi}{2}\), we have:
   \[
   \tan\left(x + \frac{\pi}{2}\right) = \frac{\tan(x) + \tan\left(\frac{\pi}{2}\right)}{1 - \tan(x)\tan\left(\frac{\pi}{2}\right)}
   \]

2. **Evaluate \(\tan\left(\frac{\pi}{2}\right)\)**:
   \[
   \tan\left(\frac{\pi}{2}\right) \text{ is undefined because it goes to infinity}.
   \]
   However, when substituting it into the formula, we treat it carefully as follows:

   \[
   \tan\left(\frac{\pi}{2}\right) = \infty
   \]

   Substituting \(\infty\) in, the expression simplifies to:

   \[
   \frac{\tan(x) + \infty}{1 - \tan(x)\cdot\infty}
   \]

3. **Simplifying the expression**:
   As \(\infty\) is involved, the whole fraction simplifies. The numerator becomes dominated by \(\infty\), and the denominator becomes very negative as \(\tan(x) \cdot \infty \) approaches infinity negatively:
   \[
   \tan\left(x + \frac{\pi}{2}\right) = -\frac{1}{\tan(x)} = -\cot(x)
   \]

Thus, the identity is verified:

\[
\tan \left( x + \frac{\pi}{2} \right) = -\cot(x)
\]

### Conclusion:
The given identity is correct.

Would you like more details or have any further questions?

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Here are 5 related questions to expand on the concept:
1. What is the sine addition formula and how is it used?
2. Can you prove the identity \(\cot(x + \frac{\pi}{2}) = -\tan(x)\)?
3. What is the relationship between tangent and cotangent functions?
4. How does the behavior of trigonometric functions change with phase shifts like \(\frac{\pi}{2}\)?
5. What are the key differences between trigonometric and inverse trigonometric identities?

**Tip:** Memorizing the basic trigonometric identities, such as tangent and cotangent relationships, can help simplify more complex problems!

Verify the identity: tan(x + π/2) = -cot(x)

Verify the trigonometric identity tan(x + π/2) = -cot(x) with a detailed step-by-step solution using the tangent addition formula and cotangent relationship. Explore key trigonometric concepts including phase shifts and tangent-cotangent identities. Suitable for students in Grades 10-12.

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